Team:Manchester/Model

We wanted a way to determine the number of our sensor bacteria required in the starter culture of a given cheese in order to ensure an accurate detection of Listeria contamination. To do this we modelled the rate of up-regulation of the agr operon in Listeria and predicted the responsiveness of our sensor.

This model is based on equations derived from https://link.springer.com/content/pdf/10.1007%2Fs00285-009-0291-6.pdf . We assumed that equations that govern the up-regulation of the agr operon in Staphylococcus would be similar to those applicable in Listeria, and so we used symbiology functions to create a step-by-step sequence of agr regulation based on the Staphylococcus template. Each step was defined as a reaction, and each reaction was governed by a kinetic law, i.e. a differential equation defining that step in the regulatory process.

Each reaction and its associated equation are shown below, see the end of this article for a table defining each variable AND IT'S UNITS.

Two compartments were generated in symbiology, one for Listeria (called cell), and one for our sensing bacteria (called sensor).

For our Listeria compartment we modelled the following reactions:

1. DNA -> mRNA -> AgrA, AgrB, AgrC, AgrD

The above image shows the equations used to model transcription and translation of the agr operon. Whilst transcription of the agr components was assumed to occur at an equal rate, given that they are all components of the same operon regulated by one upstream promoter, each component of the agr system was assigned a unique rate equation to define its translation rate, and these are given below;

Highlighted here, in blue, is the equation used to describe the agr operon transcription. The equation format is as follow:

Note: the comma in the highlighted section represents an equals sign, and this format is used for all subsequent equations. All terms are defined in an Abbreviation's table (see below).

agrB translation

agrD translation

Highlighted here, in blue, is the equation used to describe the agrA translation rate.

Highlighted here, in blue, is the equation used to describe the agrC translation rate.

2. Rate of AgrB integration into the membrane after AIP stimulus (TB)

3. Rate of AgrD association with transmembrane B after AIP stimulus

4.Rate of AgrC integration into the membrane after AIP stimulus

5. Rate of increase in the extracellular AIP concentration

6. AIP interaction with transmembrane AgrC

7. Rate of AgrA phosphorylation after AIP stimulus

8. Rate of interaction of Phosphorylated AgrA with the promoter for the agr operon (PII)

For the sensor, we used the following additional considerations

1. The movement of AIP through the cheese and subsequent sensor AgrC binding

Using a modified Stokes-Einstein equation, we aimed to estimate the rate of diffusion through cheese, depending on a number of important biophysical cheese properties. If we assume cheese can be represented as a porous matrix, in which the “pores” are fairly large (i.e. there is space for molecules to diffuse), then we can use Fick’s diffusion to describe mass transfer between these pores. This can be done by multiplying the diffusion coefficient (calculated using the Stokes-Einstein equation) by x/y (where x is equal to the tortuosity and y is equal to the porosity of the medium). The tortuosity of a medium (in our case cheese) can be described as the difficulty a molecule encounters when trying to move through the medium, whereas the porosity can be described as the ease a molecule encounters when moving through a medium. We, therefore, decided to assign the value of x (tortuosity) as a variable describing the total amount of fat, protein, and salt within the cheese (in other words, the dry mass of the cheese), as these are all the kind of molecules that AIP would encounter as obstacles to free diffusion, and thus they represent “difficulty”. For the value of y (porosity), we assigned a variable describing the water content in the cheese (in other words, the total weight of the cheese minus its dry mass), as we theorised AIP would encounter less difficulty moving through a cheese with a higher water content.

The resulting equation is as follows:

2. AgrA phosphorylation by our sensor AgrC upon AIP binding

3. Phosphorylated AgrA stimulated transcription of amilCP

Results and Discussion

Above is an image of the graph generated when our model is simulated in MatLab. Each line corresponds to an important component in the mechanism of agr upregulation in Listeria. For the proof-of-concept simulations random values were assigned to the parameters. However, so model was set up in such a way that we could have used an ensemble modelling approach by collecting possible values from sources like Bionumbers and creating a probability distribution for each parameter. These distributions could then have been used to conduct a Bayesian analysis for plausible predictions, including confidence intervals. We would have also liked to create a user input function, to allow a user to define the properties of their specific cheese. Most importantly, cheese types vary widely in water content, and our discussions with cheese makers had shown us that this is a very important consideration for Listeria detection. By exploring the effect of different biophysical condition, our model would thus allow us to determine an accurate prediction of the concentration of our sensor bacteria required for reliable Listeria detection in a user-specific starting culture. This again is an important feature of our project, requested by cheese makers in our stakeholder discussions with various companies (read more in our Human practices page).